Schauder’s Conjecture (i.eėvery compact convex set in a Hausdorff topological vector space has the f.p.p.) is reduced to the search for fixed points of suitable multivalued maps in finite dimensional spaces.

Using the Haar-Kármán principle, approximate solutions of the basic boundary value problems are proposed and studied, which consist of piecewise linear stress fields on composite triangles. The torsion problem is solved in an analogous manner. Some convergence results are proven.

Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $\mathrm{id}+K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0\le q\le n$. Then: If $V\to X$ is a holomorphic vector bundle (of finite rank) with $H^q(X,V)=0$, then $dim{H}^q(X,V\otimes E)\<\infty$. In particular, if $dim{H}^q(X,𝒪)=0$, then $dim{H}^q(X,E)\<\infty$.

For a $C^1$-function $f$ on the unit ball $𝔹\subset {ℂ}^n$ we define the Bloch norm by ${\parallel f\parallel }_{𝔅}=sup\parallel \tilde{d}f\parallel ,$ where $\tilde{d}f$ is the invariant derivative of $f,$ and then show that $${\parallel f\parallel }_{𝔅}=\underset{\genfrac{}{}{0pt}{}{z,w\in 𝔹}{z\ne w}}{sup}{(1-|z|}^2{)}^{1/2}{(1-|w|}^2{)}^{1/2}\frac{\left|f\right(z)-f(w\left)\right|}{|w-P_{w}z-s_w{Q}_{w}z|}.$$

Let Γ be a closed set in ${ℝ}^n$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants $c_1>0$ and $c_2>0$ such that $c_1{r}^d\le \mu \left(B(x,r)\right)\le c_2{r}^d$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces $L_p\left(\Gamma \right)$, 0 < p ≤ ∞, with respect to...